For the present case, one needs to evaluate the hyperdeterminant for tripartite qutrit states. One important properties of the hyperdeterminant is that, under invertible local operations \(|\tilde{\psi }\rangle \equiv \mathbf {L}_1\otimes \mathbf {L}_2\otimes \mathbf {L}_3|\psi \rangle \) , it transforms with a determinantal factor, \(\operatorname{Det}(|\tilde{\psi }\rangle )=\prod _{i=1}^3(\det (\mathbf {L}_i))^2\operatorname{Det}(|\psi \rangle )\) , where \(\mathbf {L}_i\)  (\(i=1,2,3\) ) are invertible \(3\times 3\)  matrices with complex entries [1]}, and \(|\psi \rangle \)  is an arbitrary tripartite qutrit state. When \(\mathbf {L}_i\)  are special linear transformations of degree 3, i.e., three-by-three matrices of determinant 1, the hyperdeterminant becomes an invariant under SL\((3,\mathbb {C})^{\otimes 3}\)  transformations: \(\operatorname{Det}(|\tilde{\psi }\rangle )=\operatorname{Det}(|\psi \rangle )\) . Dür showed that if two pure states can be obtained from the other by means of stochastic local operations and classical communications (SLOCC), they have the same kind of entanglement [2]}. They proved that two pure states are equivalent under SLOCC if they can be related by invertible local transformations. In other words, the hyperdeterminant is an SLOCC invariant [2]}.
